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I was trying to understand the behavior of scalar field in the vicinity of black hole and I set up the Klein-Gordon equation in the background of Schwarzschild metric. But I am having difficulty in solving this equation exactly. Also, I could only find asymptotic solutions in the literature.

Could any one please help me on this matter.

Thanks in advance.

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I understand that you are having difficulties, the resulting differential equation is not trivial:

$$r(r-1)y''(r)+(2 r-1)y'(r)-r^2 q^2 y(r)=0 $$

(where q is a constant). Nevertheless, you can analyze it numerically using Mathematica or Matlab.

But the problem is already solved: asymptotic expressions in terms of Whittaker functions were found by Rowan and Stephenson in 1977 (See J. Phys. A: Math. Gen. Vol 10, n. 1, pp. 15-23). A more general solutions has recently been found by Bezerra et al. (see arXiv:1312.4823) in terms of Confluent Heun functions. They solve the more complicated Kerr-Newman metric, for a black hole carrying charge and angular momentum, but I think you could apply this solution to the Schwarzschild metric by taking $Q=0$ and $L=0$ in their equations.

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